x(x+1)=159

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Solution for x(x+1)=159 equation:


Simplifying
x(x + 1) = 159

Reorder the terms:
x(1 + x) = 159
(1 * x + x * x) = 159
(1x + x2) = 159

Solving
1x + x2 = 159

Solving for variable 'x'.

Reorder the terms:
-159 + 1x + x2 = 159 + -159

Combine like terms: 159 + -159 = 0
-159 + 1x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '159' to each side of the equation.
-159 + 1x + 159 + x2 = 0 + 159

Reorder the terms:
-159 + 159 + 1x + x2 = 0 + 159

Combine like terms: -159 + 159 = 0
0 + 1x + x2 = 0 + 159
1x + x2 = 0 + 159

Combine like terms: 0 + 159 = 159
1x + x2 = 159

The x term is 1x.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
x + 0.25 + x2 = 159 + 0.25

Reorder the terms:
0.25 + x + x2 = 159 + 0.25

Combine like terms: 159 + 0.25 = 159.25
0.25 + x + x2 = 159.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 159.25

Calculate the square root of the right side: 12.619429464

Break this problem into two subproblems by setting 
(x + 0.5) equal to 12.619429464 and -12.619429464.

Subproblem 1

x + 0.5 = 12.619429464 Simplifying x + 0.5 = 12.619429464 Reorder the terms: 0.5 + x = 12.619429464 Solving 0.5 + x = 12.619429464 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = 12.619429464 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 12.619429464 + -0.5 x = 12.619429464 + -0.5 Combine like terms: 12.619429464 + -0.5 = 12.119429464 x = 12.119429464 Simplifying x = 12.119429464

Subproblem 2

x + 0.5 = -12.619429464 Simplifying x + 0.5 = -12.619429464 Reorder the terms: 0.5 + x = -12.619429464 Solving 0.5 + x = -12.619429464 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = -12.619429464 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -12.619429464 + -0.5 x = -12.619429464 + -0.5 Combine like terms: -12.619429464 + -0.5 = -13.119429464 x = -13.119429464 Simplifying x = -13.119429464

Solution

The solution to the problem is based on the solutions from the subproblems. x = {12.119429464, -13.119429464}

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